Homogenization of weakly coercive integral functionals in three-dimensional elasticity

TL;DR

This paper proves that in three-dimensional elasticity, homogenized tensors derived from weakly coercive energies remain strongly elliptic, extending previous two-dimensional results and analyzing the effects of lamination on ellipticity loss.

Contribution

It establishes that weak coercivity conditions still lead to strongly elliptic homogenized tensors in 3D elasticity, broadening understanding beyond 2D cases.

Findings

Homogenized tensor remains strongly elliptic under weak coercivity.

Loss of strong ellipticity can occur via rank-two lamination.

Results extend 2D homogenization theory to 3D elasticity.

Abstract

This paper deals with the homogenization through $\Gamma$-convergence of weakly coercive integral energies with the oscillating density $\mathbb{L}(x/\epsilon)\nabla v : \nabla v$ in three-dimensional elasticity. The energies are weakly coercive in the sense where the classical functional coercivity satisfied by the periodic tensor L (using smooth test functions v with compact support in $\mathbb{R}^3$) which reads as $\Lambda(\mathbb{L}) >0$, is replaced by the relaxed condition $\Lambda(\mathbb{L}) \ge 0$. Surprisingly, we prove that contrary to the two-dimensional case of [2] which seems a priori more constrained, the homogenized tensor $\mathbb{L}^0$ remains strongly elliptic, or equivalently $\Lambda(\mathbb{L}^0) >0$, for any tensor $\mathbb{L} = \mathbb{L}(y_1)$ satisfying $\mathbb{L}(y)M : M + D : {\rm Cof}(M) \ge 0$, a.e. $y \in \mathbb{R}^3$, $\forall M \in \mathbb{R}^{3\times 3}$, for some matrix $D \in \mathbb{R}^{3\times3}$ (which implies $\Lambda(\mathbb{L}) \ge 0$), and the periodic functional coercivity (using smooth test functions $v$ with periodic gradients) which reads as $\Lambda_{\rm per}(\mathbb{L})>0$. Moreover, we derive the loss of strong ellipticity for the homogenized tensor using a rank-two lamination, which justifies by $\Gamma$-convergence the formal procedure of [8].